--- title: "Bowditch balancing of link traverse XABCY" subject: "Surveying II" chapter: 1 key: "semester_4/surveying_ii:bowditch-link-traverse-xabcy" source: https://www.bctengineer.com/q/semester_4/surveying_ii/bowditch-link-traverse-xabcy site: BCT Engineering --- # Bowditch balancing of link traverse XABCY Surveying II · Chapter 1: Traversing - Asked in: 2079 Jestha, 10 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/surveying_ii/bowditch-link-traverse-xabcy - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-1 ## Question Balance the coordinates of a link traverse XABCY using Bowditch's rule. The coordinates of stations X and Y are (562.510 N, 175.250 E) and (443.610 N, 443.610 E) respectively and permissible closing error is 1:500. Other observed data are given below: | Lines | XA | AB | BC | CY | |---|---|---|---|---| | Length (m) | 120.00 | 111.50 | 132.40 | 97.60 | | Bearings | 135°00' | 119°30' | 175°00' | 77°30' | ## Short answer Compute latitudes and departures from given lengths and bearings. Misclosure from known coordinates is 133.196 m (1:3.5), exceeding permissible 1:500. For adjustment, apply Bowditch correction C = -error × (line length / ΣL). Adjusted coordinates: A (511.884 N, 254.808 E), B (488.781 N, 346.933 E), C (394.648 N, 352.630 E). ## Answer Method: compute latitudes and departures from the given WCBs, find the closing error against the known coordinates of Y, check it against the permissible limit, then distribute it by Bowditch's rule and compute the adjusted coordinates. #### Step 1: Latitudes and departures $$ \text{Lat}=L\cos\theta,\qquad \text{Dep}=L\sin\theta $$ #### Step 2: Closing error Required change in coordinates from X to Y: $$ \begin{aligned} \Delta N_{true}&=443.610-562.510=-118.900\ \text{m}\\ \Delta E_{true}&=443.610-175.250=268.360\ \text{m} \end{aligned} $$ Computed: $\Sigma \text{Lat}=-250.530$ m, $\Sigma \text{Dep}=288.723$ m. $$ \begin{aligned} e_N&=-250.530-(-118.900)=-131.630\ \text{m}\\ e_E&=288.723-(268.360)=20.363\ \text{m}\\ e&=\sqrt{e_N^2+e_E^2}=133.196\ \text{m} \end{aligned} $$ Perimeter (total length) $\Sigma L=461.50$ m, so the relative error is $e/\Sigma L = 1:3.5$. The permissible closing error is 1:500, i.e. at most 0.923 m, but the error here is 133.196 m (about 1:3.5). The given coordinates of Y and the field data do not agree, so in a real survey this traverse would be rejected and re-measured. The Bowditch adjustment is shown below only to complete the procedure. #### Step 3: Bowditch correction $$ C_{lat}=-e_N\frac{L}{\Sigma L},\qquad C_{dep}=-e_E\frac{L}{\Sigma L} $$ | Line | Length (m) | WCB | Latitude | Departure | Corr. lat | Corr. dep | Adj. lat | Adj. dep | |---|---|---|---|---|---|---|---|---| | XA | 120.00 | 135°00' | -84.853 | +84.853 | +34.227 | -5.295 | -50.626 | +79.558 | | AB | 111.50 | 119°30' | -54.905 | +97.045 | +31.802 | -4.920 | -23.103 | +92.125 | | BC | 132.40 | 175°00' | -131.896 | +11.539 | +37.763 | -5.842 | -94.133 | +5.697 | | CY | 97.60 | 77°30' | +21.125 | +95.286 | +27.838 | -4.307 | +48.962 | +90.980 | | **Sum** | 461.50 | | -250.530 | +288.723 | +131.630 | -20.363 | -118.900 | +268.360 | #### Step 4: Adjusted coordinates Start from X and add the adjusted latitude and departure of each line: | Station | Northing (m) | Easting (m) | |---|---|---| | X | 562.510 | 175.250 | | A | 511.884 | 254.808 | | B | 488.781 | 346.933 | | C | 394.648 | 352.630 | | Y | 443.610 | 443.610 | Check: the computed coordinates of Y agree with the given Y (443.610 N, 443.610 E). **Answer:** adjusted coordinates A (511.884 N, 254.808 E), B (488.781 N, 346.933 E), C (394.648 N, 352.630 E); closing error 133.196 m (1:3.5). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In a link traverse XABCY, what do the computed SigmaLat and SigmaDep have to be compared with? A. Zero B. The difference in coordinates from X to Y (N_Y - N_X, E_Y - E_X) C. The perimeter of the traverse D. The sum of the given bearings Answer: B. A link traverse does not return to the start, so closure is judged against the known coordinate difference between the two end stations. (Key point: Required coordinate change) 2. Computed SigmaLat = -250.530 m and the required change is -118.900 m. What is e_N? A. -131.630 m B. -369.430 m C. +131.630 m D. -118.900 m Answer: A. e_N = computed - true = -250.530 - (-118.900) = -131.630 m. (Key point: Closing error) 3. Permissible closing error is 1:500 and total length is 461.5 m. What does the result e = 133.196 m mean? A. It is within limit, so the traverse is accepted B. The limit is 133 m so it just passes C. It far exceeds the permissible 0.923 m, so the traverse would be rejected and re-measured D. The permissible error should be ignored for link traverses Answer: C. 461.5/500 = 0.923 m is the allowed error; 133.196 m is about 1:3.5, so the data and given Y do not agree. (Key point: Acceptance) 4. In Bowditch's rule which line gets the largest correction and with what sign here? A. The shortest line CY, with negative latitude correction B. All lines equally C. The longest line BC, with negative latitude correction D. The longest line BC, with positive latitude correction (opposite to e_N) Answer: D. C_lat = -e_N x L/SigmaL, proportional to length; e_N is negative so corrections are positive (+37.763 for BC). (Key point: Bowditch correction) ## Close variants set in other exams - [Bowditch link traverse XABCY (X 1162.510 N)](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Abowditch-link-traverse-xabcy-1162) — Surveying II, chapter 1, asked 2× ## Related in Surveying II - [Bowditch link traverse XABCY (X 1877.51 N)](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Abowditch-link-traverse-xabcy-1877) — Surveying II, chapter 1